Oscillation Criteria for some Semi-Linear Emden–Fowler ODE

Abstract

Inspired from earlier works on oscillation criteria for semi-linear elliptic equations, we pinpoint here some straightforward and easy oscillation criteria for Emden–Fowler differential equations. We find out that for  α ≥ 0, the equation

$$\displaystyle{\;[\vert y'\vert ^{\alpha -1}y']' + f(t,y) = 0\;}$$

is oscillatory if for some  m, T > 0  and $\;\beta \in [1,\;\alpha ]\quad \exists q \in C([T,\;\infty ),\;(m,\;\infty ))\;$ such that

$$\displaystyle{\forall t > T\;\text{ and}\;\forall s \in \mathbb{R},\qquad f(t,s) \geq q(t)\vert s\vert ^{\beta -1}s.}$$

The main tools for our investigation are some version of Picone identities and comparison methods. We are considering equations of the type

$$\displaystyle{\left \{\begin{array}{@{}l@{\quad }l@{}} (i)\quad \bigg\{\phi (y')\bigg\}' +\varPsi (t,y,y') = 0 \quad \\ (ii)\quad \mbox{ where $\forall S \in \mathbb{R}$ and some $\alpha \geq 0$}\quad \phi (S):=\phi _{\alpha }(S) = \vert S\vert ^{\alpha -1}S;\quad \\ (iii)\quad \varPsi \in C(\mathbb{R}^{3},\; \mathbb{R}). \quad \\ \quad \end{array} \right.}$$

Usually equations in these contexts have the form

$$\displaystyle{\bigg\{a(t)\phi (y')\bigg\}' +\varPsi (t,y,y') = 0}$$

where for some $\;t_{0} \geq 0,\quad \;a \in C^{1}([t_{0},\;\infty ))\;$ is strictly positive with  a′ ≥ 0. Because of these conditions on a, in regard of oscillatory character, that equation is equivalent to

$$\displaystyle{\bigg\{\phi (y')\bigg\}' + \dfrac{a'(t)} {a(t)}\phi (y') + \dfrac{\varPsi (t,y,y')} {a(t)} = 0.}$$

This is the reason why we take a(t) ≡ 1 in our study and extend the investigation to the equations with damping terms, ϕ(y′), say. We set the following hypotheses: (H): the function Ψ has the form

• (H1) Ψ(t, u, u′): = f(t, u)  where $\forall t \in \mathbb{R}\;$ and u ≠ 0, uf(t, u) > 0;

• (H2) $\varPsi (t,u,u'):= g(t,u') + f(t,u)\;$ which f as in (H1) and $g \in C(\mathbb{R}^{2},\; \mathbb{R})$.